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## Episode 64 - Pamela Harris and Aris Winger

## Manage episode 287107075 series 1516226

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. We need a better tagline, but I'm not going to come up with one today. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And I think that our guests might be able to help us with that tagline. But we'll get to that in a moment because I have to share with you a big kitchen win I had recently.

KK: Okay.

EL: Which is that that I successfully worked with phyllo dough! It was really exciting. I made these little pie pocket things with a potato and olive filling. It was so good. And the phyllo dough didn't make me want to tear out my hair. It was just like, best day ever.

KK: Did you make it from scratch?

EL: No, I mean, I bought frozen phyllo dough.

KK: Okay, all right.

EL: Yeah, yeah, I’m not at that level.

KK: I’ve never worked with that stuff. Although my son and I made made gyoza last month, which, again, you know, that that's a lot of work to because you start folding up these dumplings, and you know. They’re fantastic. It's much better. So, yeah, enough. Now I'm getting hungry. Okay. It's mid afternoon. It's not time for supper yet. So today we have we have a twofer today. This is this is going to be great, great fun. It's like a battle royale going here. This will be so much fun. So today we are joined by Pamela Harris and Aris winger. And why don't you guys introduce yourself? Let's start with Pamela.

Pamela Harris: Hi, everyone. I like how we're on Zoom, and so I get to wave. But that’s really only to the people on the call. So for those listening, imagine that I waved at you. So I am super excited to be here with you all today. I'm an associate professor of mathematics at Williams College. And I have gotten the pleasure to work with Dr. Aris Winger on a variety of projects, but I'll let him introduce himself too.

Aris Winger: Hey everybody, I’m Aris Winger. I'm assistant professor at Georgia Gwinnett College. I've been here for a few years now. Yeah, no, we, Pamela and I have been all over the place together. I've been the honored one, to just be her sidekick on a lot of things.

PH: Ha, ha, stop that!

EL: So we're very excited to have you here. So you've worked on several things together. The reason that I thought it would be great to have you on is that one of the things is a podcast called Mathematically Uncensored. And it's a really nice podcast. And I think it has a fantastic tagline. I was telling Aris earlier that it just made me very jealous. So we've we've never quite gotten, like, this snappy tagline. So tell us what your podcast tagline is. And a little bit about the podcast.

PH: Maybe I can do the tagline. So our tagline is “Where our talk is real and complex, but never discrete.”

AW: Yeah, that's right. That is the tagline. And yeah, it's a good one. And sometimes I have to come back to it time and again to remember, so that we live up to that during the podcast. We're taping the podcast later today, actually. And so it should be out on Wednesday. So yeah, the show is about really creating a space for people of color in the mathematical sciences and in mathematics in general, I think. And so one of the ways—I think for us the only way that can happen—is we have to start having hard conversations. Right. And so a realization that comfort and staying on the surface level of our discussions doesn't allow for us to have the true visibility that all people in mathematics should have. And so for too long, we've been talking surface-level and saying, “Oh, we have diversity issues. Oh, we should work harder on inclusion.” No, actually, people are suffering. No, actually, here's our opinion. And stop talking about us; start talking to us. So it really is a space where we're just like, you know what, screw it. Let us say what we think needs to be said. Listen to us. Listen to people who look like us. And yeah it’s hard. It's hard to do the podcast sometimes because when you go deeper and start to talk about harder topics, then there are risks that come with that. Pamela and I, week after week, say, “Oh, I don't know if I really should have said that.” But ,you know, it's what needs to be said, because we're not doing it just for us. We're doing it to model what what needs to happen from everybody in this discipline, to really say the things that need to be said.

KK: Have you gotten negative feedback? I hope not.

AW: Yeah, that’s a good question. So I mean, I think that the emails we've gotten are have been great and supportive. But I think, so for me, I'm expecting no one to say — I’m expecting the usual game as it is, right, that people aren't going to say anything, but of course there's going to be backlash when you start saying things that go against white privilege and go against the current power structures. You know, I'm expecting to be fired this year.

PH: Yeah, those are the conversations that we have constantly — that we’re having on the podcast are things that Aris and I are having conversations about privately. And so part of what's been really eye-opening for me in terms of doing a podcast is that I forget people are listening. There are times Aris and I are having just a conversation, and I forget we're recording. And I say things that I normally would censor. If I were in a mixed crowd, if I were in a department meeting, if I were at a committee meeting for, you know, X organization. And I think it's not so much that we would receive an email that says, “Hey, you shouldn't have said X, Y, and Z,” it’s that we are actually getting targeted. For example, I was just virtually visiting Purdue University giving a talk about a book that Aris and I wrote, supporting students of color. And accidentally, the link got shared to the wrong people. And all of a sudden, I'm getting Zoom-bombed at a conversation. That's targeted, right? So those are the kinds of things that we are experiencing as people of color, and we have to have conversations about how are we ensuring that this isn't the experience when you bring a Black or brown mathematician to talk virtually at your colloquium. And if we're not talking about that, then no one is talking about that, because people are trying to hide their dirty laundry. Purdue University is not putting out an email to their alumni saying, “By the way, we invited Pamela Harris to show up and talk about how we best support students of color. And then we got Zoom-bombed, and somebody was writing the N-word and saying f BLM.” Right? Like, that's not happening.

AW: Yes. Wait, they didn't say anything about it?

PH: Well, they're actively doing things about it. But you know they're not putting out the message.

AW: Right. So then it gets sanitized, right? So a traumatic attack gets sanitized to be something else. There are two things about the podcast that Pamela and I, and the Center for Minorities in the Mathematical Sciences, really are trying to work with is making sure that we call out these things, but then not to center it, right, because the the podcast itself is supposed to be about our experiences. But a lot of ways there's a significant part of our experiences that is tied to having to continuously fight against this type of oppression against us.

EL: Yeah. And I think it's really important to have that. And it's so important that it decenters— I think I was listening to an episode recently where you talked about the white gaze and what you have to deal with all the time in trying to present things to a majority white audience. And I think it's really important for us white people to listen to this and realize that not everything is about and for us. And I mean, there are so many things in life where this is true: movies, TV shows, books and stuff. And yeah, I think it's great that that your voices are there and having these conversations, and I think that people should listen to your podcast.

AW: I appreciate that. Yeah. Because it requires a deep interrogation, a self-interrogation by white people to really deal with the feelings. Let me just step back and give the usual disclaimer. Everybody's nice. Everybody's good. Nobody's mean. Nobody is a bad person. Let me just say all that to get that out of the way, right? But what we're talking about is that when I say something on the show, when Pamela says something on the show and you get this feeling like “Wow, that doesn't feel good to me,” then you need to take some time and figure out why it is that you're feeling this way. And it's tied to your privilege, something that you need to interrogate, and it will make you a better person and for everybody.

KK: I don't know. I can't wrap my head around people, like, Zoom-bombing. This is nothing that would ever come to my mind. “You know what, I'm going to go Zoom-bomb this person.” I just…

EL: Well, I mean, it’s just a bad way to spend your time, but not everyone has the same time priorities.

AW: Well, no. So I think that's a great question. And let me just say that it that's how deep and pervasive it is in people, right, that people grow up and have this experience of being raised by other people who have ingrained within them that it is fundamentally, and in some sense, it just burns their soul to have somebody who does not look like them, have someone who is “lesser” than them take the center stage, be deemed the expert. And so again, I'm not calling these people bad. But there is something within some of us that says — and it’s called white supremacy, by the way — that we all have, that we all have to fight, that is so ingrained in some people that they feel compelled to do it. And so they, again, no one's going to fix that for them. And the person who did this to Pamela has it in spades, right? And so when we say that, so I think too often we make it an intellectual exercise, right? We say that it just makes no sense. Right? It doesn't make any sense because white supremacy makes absolutely no sense. But it is a thing. And it's there. And that's what it is, right? So I've been working a lot on calling, naming things so that we don't get confused, because as long as we don't name it, then it just gets to be out there. Like, “Oh, I don't understand.” We understand this exactly. It's called white supremacy. And we need to fight it in our discipline, and across the board.

PH: And it doesn't always just show its face via Zoom-bombing with the N-word in the chat, right? It shows up with who you invite to your podcast. It shows up with who's winning awards from our big national organizations. It shows up with who gets tenure, who even lands into a tenure track position, who even gets to go into graduate school, who actually majors as a mathematician, who actually goes to college, who actually graduates high school, who actually gets told that they're a mathematician. Right? So this is showing its ugly head in very visual ways that we all feel a huge sense of, “Oh, no, this is terrible. I'm sorry, this happened to you.” But the truth is that white supremacy is in everything within the mathematical sciences. And so you know, we got to pull it at its root, my friends. At its root!

AW: Yes.

PH: So this was just one way in which it showed itself, but I want to make it clear that it is pervasive.

KK: Sure. Right.

EL: So what I love about hosting this podcast is that we get to know both people and their math and their relationship to their math. And so we're gonna pivot a little bit now, maybe pivot a lot now, and say, Okay, what are your favorite theorems? And, yeah, I don't know who wants to go first. But, yeah, what's your favorite theorem?

KK: Yeah, let’s hear it.

PH: I’ll do it. I’ll go first. I always like hearing Aris talk. So I'm just like “Aris, go,” right? But no, I’m going to take the lead today. Alright, so I wanted to tell you about this theorem called Zeckendorf’s theorem. I don't know if you know about it.

KK: I do not.

PH: And it goes like this. So start with the Fibonacci numbers without the repeated 1. So 1, 2, and then start adding the previous two, so 3-5-8, and so on. Alright. So if you start with that sequence, his theorem says the following, if you give me any positive integer N, I can write it uniquely as a sum of non-consecutive Fibonacci numbers.

AW: Oh, wow!

EL: Uniquely?

PH: Yes. And this is why you need to get rid of the 1, 1. Because otherwise you have a choice. But yeah. So it's hard to do off the top of my head, because I'm not someone who, like, holds numbers. But say, for example, we wanted to do 20. Maybe we wanted to write the number 20 as a sum of Fibonacci numbers that are not consecutive. So what would you do? You would find the largest Fibonacci number that fits inside of 20. So in this case, it would be 13.

AW: Yeah.

PH: 13 fits in there. Okay, so we subtract 13. We're left with 7. Repeat the pattern.

KK: Ah, five and two.

KK: Five and two! They're non consecutive.

KK: Okay.

PH: Yeah.

AW: Wow!

PH: Three is in between them, and eight is in between the others. And so you can do this uniquely. And so this is using what's known as the greedy algorithm because you just do that process that I just said, and it terminates because you started with a finite number.

KK: Sure.

PH: And so the the proof, of course, there's the, you know, “Can you do it?” but then “Can you do it uniquely?” So the thing that you would do there is assume that you have two different ways of writing it, each of which uses non-consecutive, and then you would argue that they end up being exactly the same thing. So that, in fact, they use the same number of Fibonacci numbers and that those numbers are actually the exact same.

KK: Sure, okay.

EL: Yeah. Like I'm trying to figure out — and I don't, I also am not super great at working with numbers in my head just on the fly. But yeah, I'm trying to figure out, like, what would have gone wrong if I had picked eight instead of 13 to start with, or something? And I feel like that will help me understand, but I probably need to go sit quietly by myself and think about it. Because there’s a little pressure.

PH: Yeah, it's a little subtle. And it might be that you don't get big enough, you end up having to repeat something.

EL: Yeah, I feel like there's not enough left below eight to get me there without being consecutive.

PH: Yeah. Right.

AW: Right. Because you’ve got to get 12. Yeah, yes. Yeah.

KK: Yeah, it makes sense, right? Like, I guess, you know, if you pick the largest one less than your number, then it's more than halfway there. That's sort of the point, right? So that's how you prove it terminates, but also the the non-uniqueness, the non-uniqueness seems like the hard part to me somehow, but also the non-consecutive. Wait a minute, I don't know, which is.

AW: Well it sounds hard, period.

KK: Yeah. I like this theorem. This is good. What attracts you about this theorem? What gets you there?

PH: So I, in part of my dissertation, I found a new place where the Fibonacci numbers showed up. And so once you find Fibonacci numbers somewhere new, I was like, what else is known about these beautiful numbers? And so this was one of those results that I found, you know, just kind of looking at the literature. And then I later on started doing some research generalizing this theorem. So meaning, in what other ways could you create a sequence of numbers that allows you to uniquely write any positive integer in this kind of flavor, right, that you don't use things consecutively, and consecutively, really, in quotes, because you can define that differently. And so it led me to new avenues of research that then I got to do. It was the first few research projects with some of my undergraduate students at the Military Academy. And then I learned through them — they looked him up — that he actually came up with this theorem while he was a prisoner of war.

AW: Oh, wow!

PH: This is when Zeckendorf worked on this theorem. And to me, this was really surprising that, you know, my students found this out. And then I was like, “See, mathematics, you can just take it anywhere.” lLke this poor man was a prisoner of war, and he's proving a theorem in his cell.

KK: Well Jean Leray figured out spectral sequences in a German POW camp.

PH: I did not know that!

AW: Anything to pass the time.

KK: What else are you going to do?

EL: I mean, Messiaen composed the Quartet for the End of Time — I was about to say string quartet, but it's a quartet for a slightly different instrumentation, in a concentration camp, or a work camp. I'm not sure. But yeah, I'm always amazed that people who can do that kind of creative work in those environments, because I feel like, you know, I've been stuck in my house because of a pandemic, and I'm, like, falling apart. And my house is very comfortable. I have a comfortable life. I am not as resilient as people who are doing this. But yeah, that is such a cool theorem. I'm so glad that you said that. And I'm trying to think, like, Lucas numbers are another number sequence that are kind of built this way. And so is there anything that you can tell us about the the sequences that you were looking at, like, I don't know, does this work for Lucas numbers? I don't know if you've looked at that specifically, or did you look at ways to build sequences that would do this?

PH: Yeah. So we started from the construction point of view. So rather than give me a sequence, and then tell me how you can uniquely decompose a number into a sum of elements in that sequence, we worked backwards. So one of the research projects that we started with is what we called — there's a few of them — but one, it was a “Generacci” sequence. And so what we would do is, instead of thinking of the numbers themselves, imagine that you have buckets, an infinite number of buckets, you know, starting out the first bucket all the way to infinity, and you get to put numbers into the sequence in the following way. So you input the number 1 to begin with, because you need a number to start the sequence. And since you want to write all positive integers, well, you’ve got to start with 1 somewhere. So you stick the number 1 in the first bucket. And then you set up some system of rules for which buckets you can use to pull numbers from that then you add together to create new numbers. Well, you only have one bucket, and you only put the number 1 in it. So then you move to the next bucket. Well, okay, you want to build the number two, and you only have the number one, and as soon as you pull it from the bucket, you don't have any other numbers to use. So let's stick the number two in the second bucket. Oh, well, now I could maybe in my rule, grab a number from two buckets, and add them together to get the next number. Oh, that starts looking familiar. The third bucket will have not the number three, because you were able to build it. So what next number could you grab? Well, maybe you can stick in the 4 in there. And so by thinking of buckets, the numbers that you can fill the buckets with, that you couldn't create from grabbing numbers out of previous buckets under certain rules, you now start constructing a sequence. And provided that you very meticulously set up the rules under which you can grab numbers out of the buckets to add together to build new numbers, then you do not need to add that number into the buckets, because you've already built it.

EL: So what rules you have about the buckets will determine what goes in the buckets.

PH: Exactly, exactly. So you might say okay, maybe our buckets can contain three numbers. And you're not allowed to take numbers out of consecutive buckets, or neighboring buckets, or you must give five buckets in between. So what must go into the buckets to guarantee that you can create every single number and you can do so only uniquely? And so these are these bin decompositions of numbers. But you are working backwards. You start with all the numbers, and then you decide how you can place them in the buckets and how you can pull them from the buckets to add together. So I'm being vague on purpose, because it depends on the rules. And actually it's quite an open area of research, how do you build these sequences? You set up some some capacity to your bucket, some rules from where you can pull to add together. And the nice thing is that it's very accessible, and then it leads to really beautiful generalizations of these kinds of results like that of Zeckendorf.

EL: This is very cool.

AW: Fantastic.

EL: All right. So Aris, I feel like the gauntlet has been thrown.

KK: Yeah.

AW: Yeah, well mine is simple. This is not a competition. Yeah, no. I guess mine is influenced — I’ve been thinking about a bunch of different things, but I keep coming back to the same one, which I think is influenced by my identity as a teacher first and foremost when I think about the fundamental theorem of calculus. I just keep coming back to that one. And I don't know how many people have used this one with you on this podcast before, but for me, it hits so many of the check marks of my identity in terms of thinking about myself as a mathematician and a teacher, in the sense that for a lot of students who get to calculus, it's one of the first major, major theorems that will show up in their faces that we actually call out and say this is a theorem. And we call it fundamental, right? We don't often bring up the fundamental theorem of algebra in college algebra, right? Or in other places, or the fundamental theorem of arithmetic, right? But so it's one of these first fundamental theorems. And so it also helps to tell the story of a course, right? And so that really hits the teacher part of me where too often people in the calculus sequence spend all this time talking about derivatives, and all of a sudden, we just switch the anti-derivatives. And we don't really say why. You'll figure out in the next couple of sections, and then we start adding rectangles, and we don't say why. And so it really is, at least the way that the order of calculus has gone and in terms of how to teach it, in my experience, it really is this combination, like, oh, this is why we've been doing this. And this is the genius of relating two things. Sometimes I've gone in, and I've talked about, like I put up a sine curve, and a cosine curve, and we talk about how one of them measures the area under the curve. And then I pretend to bump my head and get amnesia. And then I'll come and say, “Oh, look, looks like we've been talking about derivatives. Right?” And they’re like, “Wait, what do you mean we're talking about derivatives?” “This is the derivative of this one.” And they’ll go, “What? We were measuring the area under the curve.” “Well, we’re also measuring the derivative, right?” This is the derivative, but this is measuring the area. And it's like, Oh, right, and so it's just one of these “aha” moments, where if people have been paying attention, it's like, oh, that's actually pretty cool, right? And then also in terms of the subject itself in relationship to high school, just really thinking about — because I get a lot of students who know all the rules, right? And they look at the anti-derivative with the integral sign and say “That's the integral.” Well, that's an anti-derivative, right?

EL: We’re not there yet.

AW: Yeah, that the anti-derivative and the integral are actually different. And so just having that conversation. And it also is a place to talk about the history of the subject and stuff like this.

EL: Yeah, I love it. And, at least for me, I feel like it's a slow burn kind of theorem. The first time you see it, you're like, “Okay, it's called the fundamental theorem of calculus. I guess some people think it's really important.” So that might be your Calculus I class. And then you see it again, maybe in an introductory real analysis class. And you're like, “Oh, there's more here.” And then you teach calculus, and you’re like, “Ohhhh!”

AW: Oh right, yes!

EL: Your brain explodes. You're like, “This is so cool!” And then your students are where you were several steps ago. And they’re like, “Okay, I guess it’s all right.”

AW: If I get the success rate of like, I've had three or four people go, “Whoa!” And it's like, okay, yeah, you're with me. And so this is out of hundreds of people.

EL: If you can get a few people that do that the first time they see it, that’s awesome.

AW: Yes. Yeah. No, it's been fulfilling for sure. And so then the proof itself, you know, it's also great, because then it culminates all of the theorems that you've been talking about beforehand. Depending on the proof, of course, but like, there's the intermediate value thereorem. There's the mean value theorem for integrals. There's unique continuity, at least in this version of it, in order for it to work. So yeah, it's great.

KK: So when you teach calculus, there's always two parts to the fundamental theorem. And so I like the one where the derivative of the integral is the function back, right? That's the fun, like for the mathematician in me, this is the fun part. Your students never remember that. Right? They always remember the other one, where we evaluate definite integrals by finding it the anti-derivative. So I was going to ask, if you had to pick one of the two, which one is your favorite?

AW: I mean, part of it is because at least the way that I've taught it, we're coming out of the mire of Riemann sums.

KK: Right.

AW: And so people have suffered through doing rectangles so much. And then I just get to say, “Oh, you don't have to do this anymore.” I mean, I've had a few students go, you know, now that we do — I always use the antiderivative of x squared on zero to 10, or the area under the curve of x squared from zero to 10. And like, sometimes I'll say, “Oh, that's 1000 over 3, right?” And then it was like, “Well, how did you do that so quickly?” We'll see. Right? But then, at the end, when I'll say, okay, and then we do another one again. And then I show how to apply the theorem, and people say, “Well, why didn't you just say that?” And then we have a great conversation there about how this isn't about the answer, that this is about a process and understanding the impact of mathematical ideas, that the theorem, as with all theorems, but this one is my favorite, is an expression of deep human intellect. And that if we reframe what theorems are, we get a chance to rehumanize mathematics. And so I think that too often in our math classes, and our math discourse, we remove the theorems from the humanity of the people who created them. And so people get deified, like Newton and Leibniz, but you know, these same people had to sit down and work hard at it and figure it out.

KK: No, it’s certainly a classic, but it is surprising how little it has come up on our podcast. It was the very first episode.

AW: Oh, okay.

KK: Yeah. Amie Wilkinson chose it. And then this will be episode 60-something.

AW: Okay. Yeah.

PH: Wow.

EL: We've talked, we've mentioned it in some other episodes. But it isn’t — I mean, there are just — I love this podcast, obviously, I keep doing it. And there are just so many types of theorems. And I love that you two picked different types. Yours, Aris, is one of these classics. Everyone who gets to a certain point in math has seen it, hopefully has appreciated it also. And Pamela, you picked one that none of us had ever heard of and made us say, “Whoa, that's so cool!” And people just have so many relationships with yours. And that's what this podcast is really about. Actually it's not about theorems. It's about human relationships with theorems and what makes humans enjoy these theorems. And so you picked two different ways that we enjoy theorems. And I just love that. So yeah.

KK: Yeah, that is what we're about here, actually. I mean, I mean, yeah, the theorem. But actually what I like most about our podcast, so let's toot our own horn here. We’re trying to humanize mathematics. I think everybody has this idea that mathematicians are a very monolithic bunch of weird people who just — well, in movies we’re always portrayed as either being insane, or just completely antisocial. And I mean, there’s some truth and every stereotype, I suppose, but we are people, and we love this thing. We think it's so cool. And sharing that with everyone is really what's so much fun.

AW: Yeah. And I think also that, for me, the theorem itself, and what it reveals, touches something that’s inside of us. There’s something about it, right? There’s the “Whoa” part that is that is indescribable and that I think really touches to our humanity. There is a eureka moment where you're just like, “Oh, I understand this now.” Or this connection is amazing, right? Yeah, it's indescribable.

KK: So we all agree these things are beautiful. So here's a question. Where do people lose this? I mean, I have a theory, but — because we've all had this experience, right? You're at a cocktail party and someone says, they find out you're a mathematician and like, oh, record scratch. I hate math. Okay.

AW: Yes, yes, yes.

PH: But I don’t think they hate math, though, Kevin.

KK: No, they don’t. Nobody hates math. Nobody hates math when they're a kid. That's exactly right. So I think when they say that they mean that the algebra caused them trouble. When x’s started showing up.

PH: I don't even think that's it.

KK: Okay. Good. Enlighten me because I want an answer to this that I can’t find.

PH: I don't think it's that people hate math or that they hate that the alphabet showed up all of a sudden in math that they hate how people have made them feel when they struggle with math. Math is an inanimate object. Math is not going out there and, like, punching people in the face. It's the way that people react to other people's math. Right? The second that you don't use the language in the way that somebody expects you to use it and you're trying to communicate properly and somebody says, “That’s not how you say it. It's not FOILing. It's called distributing!” Right? But you knew what I meant when I said FOIL the binomial!

KK: Of course I did.

PH: FOILing this gives you the middle term, blah, blah, right? So it’s again about human interactions. And if you make someone feel dumb, they'll never like what it is that they're trying to learn

AW: Amen to that. And they will conflate the two, which is what always happens.

PH: That’s exactly it!

AW: They will replace the experience with the subject itself, when in fact, they're talking about the experience. Yeah. So yeah, we've been working a lot about this in the last few years, Pamela and I and Dr. Michael Young, about when people say they hate mathematics, they’re really talking about their mathematical experience. So my immediate response to your question is just bad teaching. Let's just call it what it is.

PH: Right.

AW: I don't want to get on my podcast too early. We're recording later.

PH: We’re recording in a bit, yeah.

AW: But yeah, we're talking about people. And I say this as a loving critique of the greatest discipline in the history of people. I truly believe that, but I believe that the way we teach it, and the cultural norms we take with it, devalues people, and so I want every person who's listening to this now to then the next time they hear somebody say they hate it, look at them as an innocent person who had a bad mathematical experience. And then, because I see too often amongst my people in the community who say they hate having these conversations with people who say they hate it. And I think we need to return innocence back to that person. And say that this is not a person who hates you or even hates the subject. This is a hurt person. Yes, this is a person who has been damaged in our subject. And by the way, I go much farther than that. It's our responsibility to try and help repair that because this person is going to impact their cousin, their child, their relative, by bringing this hate of the subject, when in fact, it doesn't have anything to do with the subject.

EL: Yeah. It’s about the traumatic experiences. And actually, I think mathematicians often have a bit of a persecution complex and think this is the only place where people have this reaction. But one of my hobbies is singing, and in particular, singing with large groups of untrained people who are just singing because we love singing. And the baggage that people bring to singing is similar. I’m not saying it's entirely the same, but people have been made to feel like their voice isn't good enough.

AW: Yes.

EL: They have this trauma associated with trying to go out and do this sometimes. Obviously a lot of people love to sing and will do it in public. A lot of people love to sing at home and are scared of doing it in public because they're worried about, you know, their fourth grade music teacher, who told them to sing quieter, or whatever happened.

PH: Yes,

AW: That’s right. That's right. And the connection is similar, because what are we saying? We're saying that if you don't hit this right note, then it doesn't count. As opposed to if you don't get the answer seven, then we're not going to value you because the answer is seven, right? Because we have this obsession with the correct answer in mathematics. Right.

PH: And not only that, but also doing it fast.

AW: Yes.

PH: You and I have talked about this before, that — maybe in singing, this is different. I'm not sure. I definitely can relate to the trauma of never singing out loud in public. But is there this same sentiment that you must get it perfect the first time and pretend that it doesn't actually take you hours of training?

EL: I mean, it comes up. There’s definitely, people can feel more valued if they're quicker at picking things up than others, although, you know, it's not the same. There's no isomorphism between these two, I don't know, to bring a little silly math lingo in. But there definitely, there are a lot of similarities, and I think about this a lot, because two things I love in my life are math and singing with my friends. And, you know, I just see these relationships. But yeah, I could go on a whole rant, and I want to not do that.

AW: No, no, no, I appreciate you bringing it up.

EL: But I think it's a really interesting correspondence.

AW: And then the final one is that, you know, in the music space, what is it that we really should be trying to do, value everybody's voice? And in mathematics, we should be valuing everybody's contribution. Right? This is all we're saying. And what does each discipline look like when we value people's voices, no matter where they are on the keys? And we value everyone's contribution to trying to solve a problem.

EL: Yeah, yeah. And how can we help people, you know, grow in the way they want to? You can say, like, “Oh, I like I am not as good a sight reader as I want to be. How can I get better?” How can we help people grow in that way without feeling cut down?

AW: Yeah.

EL: Yeah, it is true for math, too. Yeah. It's just, everything is connected. Woo.

AW: Yes. But you know, we've been talking about, you know, these human relationships we all have with math. And so another part of our podcast that we love is forcing you to do make one more human connection between math and something else with the pairing. So what goes well, Pamela, with this theorem about uniquely writing the numbers in terms of the Fibonacci sequence?

PH: So I was trying to think about my favorite food, and when it was the epitome of perfection, and I came up with, okay, so if we're going to pair it with something to drink, I was like, I want to think about happy moments. Because this feels like a happy theorem. And so I want to go with some champagne.

KK: Okay.

PH: Okay, I was like, “We're gonna go fancy with it!” But then for food, I'm thinking about, oh, this is hilarious. So I went to a conference in Colombia, we visited Tayrona which is a beach in Colombia. And on the side of the beach, I paid to have ceviche, fresh ceviche. And I've never been happier eating anything in my life. And so I imagine myself learning Zeckendorf’s theorem at the beach in Tayrona in Colombia, with some champagne and the ceviche.

EL: Oh man.

AW: Wow.

PH: Beat that, Aris! Beat. That.

AW: There’s no way. So wait, so I want to make sure I understand. So is this while you're reading the proof? Or is this while you’re—

PH: This is like the gold standard. If I were to put all the, like, uniqueness of my favorite food, my favorite drink and my favorite theorem, I would put them in a location which is Tayrona in Colombia, at the beach, eating ceviche sipping on some champagne, learning Zeckendorf’s theorem.

AW: Okay.

KK: Is this the Pacific coast or the Caribbean?

PH: You’re asking questions I should know the answer to, and I believe it’s the Caribbean.

KK: Okay.

PH: Nobody Google that. [Editor’s note: I Googled that. It is the Caribbean.] I have no idea where they took me in Colombia. I just went.

KK: Sure.

EL: Yeah, that sounds so lovely as I look out of my window where there's snow and mud from some melted snow.

PH: Ditto.

AW: So I yeah, I think for the fundamental theorem of calculus, I think this is something that's just classic. Like you're just having a nice pizza and some ginger ale. You're just sitting down and you're enjoying something hopefully that everybody likes and that connects with everybody, that everybody hopefully sees that they get to get that far. So yeah, I mean, my daughter recently — I didn't realize this. She's 9. And we were talking. We visited my aunt in DC. My aunt raised me. And my daughter was much younger at that time, but then every time she thinks about going to visit, she thinks about the ginger ale that my aunt got her because that was the only time she ever got ginger ale. So she’s like, “Oh, I like your aunt, Daddy, because you know, I had ginger ale there.” And I was like, Oh, I should have ginger ale more often. So that made me think of that.

PH: That’s adorable.

EL: I can really relate to that feeling of, like, when you're a kid, something that is totally normal for someone else isn't what's normal for your family. So you think it's a super special thing.

AW: It’s amazing.

EL: I think I had this with, like, Rice-a-Roni or something at my aunt's house, and my mom didn't use Rice-a-Roni, and I was like, “Whoa, Mom, you should see if you can find Rice-a-Roni.”

PH: Amazing.

EL: She was like, “Yeah, they have Rice-a-Roni here.”

AW: Rice-a-Roni’s the best.

KK: I haven't had that in years. I should go get some.

AW: Me either. All right.

PH: That’s how you know you made it.

KK: You know what? You know, single mom and all that, and I lived on Kraft macaroni and cheese when I was a kid. And yeah, you would think I don't like it any more. But, aw man.

PH: Listen, that thing is delicious. So good.

AW: I was about to say.

EL: They know what they’re doing. Yeah. Well, that's great. And I mean, pizza is my favorite food. As great as ceviche on the beach sounds, pizza, just, when you come down to it, it's my favorite food. And so I love that you paired the fundamental theorem of calculus with my favorite food.

KK: So I'm curious, there must be a human who doesn't like pizza, but have you ever met one? I've never met one.

PH: No.

EL: I know people who don't like cheese. And cheese is not — I mean, to me cheese is essential to the pizza experience, but you can definitely do a pizza without cheese.

AW: Yeah. No, my wife also always says that for her it's about the sauce. So I think she might be a person who can get rid of the cheese if the sauce is right. Yeah.

KK: But the crust better be good too.

AW: Of course, of course. It's a full package here.

EL: But okay, so you say that, but on the other hand, I would say that bad pizza is still really good.

KK: Sure.

EL: I mean, you can have pizza that you're like, “I wish I didn't eat that.” But I have very rarely in my life encountered a slice of pizza that was like, “Oh, I wish I wish I had done something else other than eat that pizza.”

AW: It’s actually a pretty unbeatable combination, right? Tomato sauce, cheese and bread.

PH: Yeah. It kind of can't go wrong. Yeah.

KK: When I was when I was in college, there was a place in town. It was called Crusty’s Pizza, and I don't think it exists anymore. And it was decidedly awful. But we still got it because it was cheap. So we would occasionally splurge on the good pizza. But you could get a Crusty’s pie for like five bucks.

AW: Absolutely.

KK: This is dating myself. But yeah, absolutely. Always. All right, so we've got we've got theorems, we’ve got pairings. You've plugged your podcast pretty well, although you can talk about it more if you'd like. Anything else that either of you want to plug, websites, the Twitter?

EL: Yeah, but can you say a little more about the book that you mentioned?

AW: Yeah, the book is a series of dialogues that was an extension of an AMS webinar series that we gave about advocating for students of color mathematics. And so we had just decided, you know, there was so much momentum, we had hundreds of people coming every time to the four-part series. And so we were like, you know, we've gotten to a place where we've given all these talks, and then you give talks, create momentum, and then it just ends. And we're just like, you know, what, not this time. Let's create a product out of this. And so, we decided quickly to get the book together, just answering some of the unanswered questions from the webinar series. So we had the motivation, in terms of answering their questions. And yeah, we got it together. And it was an honor. So it really is just a list of our dialogues, a transcription of our dialogues, answering some of the unanswered questions from that webinar series. And so it's gotten some really good reviews, and people are using it in their departments. And so it's been fantastic so far.

PH: Yeah, I think that's that's the part that I'm really enjoying, getting the emails from people who have purchased the book. And so maybe I should say the full title, so it is Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics. And the things that I hear from folks who have purchased the book — so thank you all so much for the support — is that they didn't expect that there is part of a workbook involved in the book. So it isn't just Aris and I going back and forth at telling you things. I mean, a lot of that there is, that is part of the content. But there's also a piece about doing some pre-reflection before we start hearing some of the dialogue that we have, and then also the post part of it. So how are you going to change? And how are you going to be a better advocate for students of color in mathematics? And so it leaves the reader with really a set of tools to come back to time and time again. That's really what I see as a benefit of the book. And people are purchasing it as a department to actually hold some kind of book club and really think about what of the things that we suggest that professors implement in their department, in their classrooms, in their institutions, what they can actually do. And so the reception has been really wonderful. And I'm just super thankful that people purchase the book, and we're supporting our future work.

EL: Yeah. And can you also mention, is it minoritymath.org, the website that hosts Mathematically Uncensored?

AW: That’s correct. That's right. So yeah, that's the home of the podcast. And that's a place where we're trying to create voices for underrepresented minorities in the mathematical sciences. And so you can go there not just for the podcast, but for other content as well that centers around that experience.

KK: Okay.

EL: Fantastic. Thank you so much for joining us.

KK: Yeah.

EL: I had a blast.

PH: Thank you.

KK: This was a really good time.

EL: Yeah. Over lunch today, I'm going to be writing down numbers and writing them in terms of Fibonacci numbers. It’s great.

AW: It will be fantastic.

PH: Awesome.

AW: Thanks.

PH: Bye, everyone.

KK: Thanks, guys.

On this very special episode, we had not one but two guests, Pamela Harris from Williams College and Aris Winger from Georgia Gwinnett College, to talk about their podcast, Mathematically Uncensored, and of course their favorite theorems. Here are some links you might be interested in as you listen to the episode.

Winger's profile on Mathematically Gifted and Black

Mathematically Uncensored, the podcast they cohost

Minoritymath.org, the Center for Minorities in the Mathematical Science, a website with information and resources for people of color in mathematics

Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics, their book

Zeckendorf's theorem and a biography of Edouard Zeckendorf

Jean Leray, a French mathematician who worked on spectral sequences as a prisoner of war

Olivier Messiaen's Quartet for the End of Time, composed when he was a prisoner of war

A paper generalizing the Zeckendorf theorem by Harris and coauthors

Our episode with Amie Wilkinson, who also chose the Fundamental Theorem of Calculus, making it 2 for 2 among mathematicians with the initials AW.

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Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. We need a better tagline, but I'm not going to come up with one today. I'm Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And I think that our guests might be able to help us with that tagline. But we'll get to that in a moment because I have to share with you a big kitchen win I had recently.

KK: Okay.

EL: Which is that that I successfully worked with phyllo dough! It was really exciting. I made these little pie pocket things with a potato and olive filling. It was so good. And the phyllo dough didn't make me want to tear out my hair. It was just like, best day ever.

KK: Did you make it from scratch?

EL: No, I mean, I bought frozen phyllo dough.

KK: Okay, all right.

EL: Yeah, yeah, I’m not at that level.

KK: I’ve never worked with that stuff. Although my son and I made made gyoza last month, which, again, you know, that that's a lot of work to because you start folding up these dumplings, and you know. They’re fantastic. It's much better. So, yeah, enough. Now I'm getting hungry. Okay. It's mid afternoon. It's not time for supper yet. So today we have we have a twofer today. This is this is going to be great, great fun. It's like a battle royale going here. This will be so much fun. So today we are joined by Pamela Harris and Aris winger. And why don't you guys introduce yourself? Let's start with Pamela.

Pamela Harris: Hi, everyone. I like how we're on Zoom, and so I get to wave. But that’s really only to the people on the call. So for those listening, imagine that I waved at you. So I am super excited to be here with you all today. I'm an associate professor of mathematics at Williams College. And I have gotten the pleasure to work with Dr. Aris Winger on a variety of projects, but I'll let him introduce himself too.

Aris Winger: Hey everybody, I’m Aris Winger. I'm assistant professor at Georgia Gwinnett College. I've been here for a few years now. Yeah, no, we, Pamela and I have been all over the place together. I've been the honored one, to just be her sidekick on a lot of things.

PH: Ha, ha, stop that!

EL: So we're very excited to have you here. So you've worked on several things together. The reason that I thought it would be great to have you on is that one of the things is a podcast called Mathematically Uncensored. And it's a really nice podcast. And I think it has a fantastic tagline. I was telling Aris earlier that it just made me very jealous. So we've we've never quite gotten, like, this snappy tagline. So tell us what your podcast tagline is. And a little bit about the podcast.

PH: Maybe I can do the tagline. So our tagline is “Where our talk is real and complex, but never discrete.”

AW: Yeah, that's right. That is the tagline. And yeah, it's a good one. And sometimes I have to come back to it time and again to remember, so that we live up to that during the podcast. We're taping the podcast later today, actually. And so it should be out on Wednesday. So yeah, the show is about really creating a space for people of color in the mathematical sciences and in mathematics in general, I think. And so one of the ways—I think for us the only way that can happen—is we have to start having hard conversations. Right. And so a realization that comfort and staying on the surface level of our discussions doesn't allow for us to have the true visibility that all people in mathematics should have. And so for too long, we've been talking surface-level and saying, “Oh, we have diversity issues. Oh, we should work harder on inclusion.” No, actually, people are suffering. No, actually, here's our opinion. And stop talking about us; start talking to us. So it really is a space where we're just like, you know what, screw it. Let us say what we think needs to be said. Listen to us. Listen to people who look like us. And yeah it’s hard. It's hard to do the podcast sometimes because when you go deeper and start to talk about harder topics, then there are risks that come with that. Pamela and I, week after week, say, “Oh, I don't know if I really should have said that.” But ,you know, it's what needs to be said, because we're not doing it just for us. We're doing it to model what what needs to happen from everybody in this discipline, to really say the things that need to be said.

KK: Have you gotten negative feedback? I hope not.

AW: Yeah, that’s a good question. So I mean, I think that the emails we've gotten are have been great and supportive. But I think, so for me, I'm expecting no one to say — I’m expecting the usual game as it is, right, that people aren't going to say anything, but of course there's going to be backlash when you start saying things that go against white privilege and go against the current power structures. You know, I'm expecting to be fired this year.

PH: Yeah, those are the conversations that we have constantly — that we’re having on the podcast are things that Aris and I are having conversations about privately. And so part of what's been really eye-opening for me in terms of doing a podcast is that I forget people are listening. There are times Aris and I are having just a conversation, and I forget we're recording. And I say things that I normally would censor. If I were in a mixed crowd, if I were in a department meeting, if I were at a committee meeting for, you know, X organization. And I think it's not so much that we would receive an email that says, “Hey, you shouldn't have said X, Y, and Z,” it’s that we are actually getting targeted. For example, I was just virtually visiting Purdue University giving a talk about a book that Aris and I wrote, supporting students of color. And accidentally, the link got shared to the wrong people. And all of a sudden, I'm getting Zoom-bombed at a conversation. That's targeted, right? So those are the kinds of things that we are experiencing as people of color, and we have to have conversations about how are we ensuring that this isn't the experience when you bring a Black or brown mathematician to talk virtually at your colloquium. And if we're not talking about that, then no one is talking about that, because people are trying to hide their dirty laundry. Purdue University is not putting out an email to their alumni saying, “By the way, we invited Pamela Harris to show up and talk about how we best support students of color. And then we got Zoom-bombed, and somebody was writing the N-word and saying f BLM.” Right? Like, that's not happening.

AW: Yes. Wait, they didn't say anything about it?

PH: Well, they're actively doing things about it. But you know they're not putting out the message.

AW: Right. So then it gets sanitized, right? So a traumatic attack gets sanitized to be something else. There are two things about the podcast that Pamela and I, and the Center for Minorities in the Mathematical Sciences, really are trying to work with is making sure that we call out these things, but then not to center it, right, because the the podcast itself is supposed to be about our experiences. But a lot of ways there's a significant part of our experiences that is tied to having to continuously fight against this type of oppression against us.

EL: Yeah. And I think it's really important to have that. And it's so important that it decenters— I think I was listening to an episode recently where you talked about the white gaze and what you have to deal with all the time in trying to present things to a majority white audience. And I think it's really important for us white people to listen to this and realize that not everything is about and for us. And I mean, there are so many things in life where this is true: movies, TV shows, books and stuff. And yeah, I think it's great that that your voices are there and having these conversations, and I think that people should listen to your podcast.

AW: I appreciate that. Yeah. Because it requires a deep interrogation, a self-interrogation by white people to really deal with the feelings. Let me just step back and give the usual disclaimer. Everybody's nice. Everybody's good. Nobody's mean. Nobody is a bad person. Let me just say all that to get that out of the way, right? But what we're talking about is that when I say something on the show, when Pamela says something on the show and you get this feeling like “Wow, that doesn't feel good to me,” then you need to take some time and figure out why it is that you're feeling this way. And it's tied to your privilege, something that you need to interrogate, and it will make you a better person and for everybody.

KK: I don't know. I can't wrap my head around people, like, Zoom-bombing. This is nothing that would ever come to my mind. “You know what, I'm going to go Zoom-bomb this person.” I just…

EL: Well, I mean, it’s just a bad way to spend your time, but not everyone has the same time priorities.

AW: Well, no. So I think that's a great question. And let me just say that it that's how deep and pervasive it is in people, right, that people grow up and have this experience of being raised by other people who have ingrained within them that it is fundamentally, and in some sense, it just burns their soul to have somebody who does not look like them, have someone who is “lesser” than them take the center stage, be deemed the expert. And so again, I'm not calling these people bad. But there is something within some of us that says — and it’s called white supremacy, by the way — that we all have, that we all have to fight, that is so ingrained in some people that they feel compelled to do it. And so they, again, no one's going to fix that for them. And the person who did this to Pamela has it in spades, right? And so when we say that, so I think too often we make it an intellectual exercise, right? We say that it just makes no sense. Right? It doesn't make any sense because white supremacy makes absolutely no sense. But it is a thing. And it's there. And that's what it is, right? So I've been working a lot on calling, naming things so that we don't get confused, because as long as we don't name it, then it just gets to be out there. Like, “Oh, I don't understand.” We understand this exactly. It's called white supremacy. And we need to fight it in our discipline, and across the board.

PH: And it doesn't always just show its face via Zoom-bombing with the N-word in the chat, right? It shows up with who you invite to your podcast. It shows up with who's winning awards from our big national organizations. It shows up with who gets tenure, who even lands into a tenure track position, who even gets to go into graduate school, who actually majors as a mathematician, who actually goes to college, who actually graduates high school, who actually gets told that they're a mathematician. Right? So this is showing its ugly head in very visual ways that we all feel a huge sense of, “Oh, no, this is terrible. I'm sorry, this happened to you.” But the truth is that white supremacy is in everything within the mathematical sciences. And so you know, we got to pull it at its root, my friends. At its root!

AW: Yes.

PH: So this was just one way in which it showed itself, but I want to make it clear that it is pervasive.

KK: Sure. Right.

EL: So what I love about hosting this podcast is that we get to know both people and their math and their relationship to their math. And so we're gonna pivot a little bit now, maybe pivot a lot now, and say, Okay, what are your favorite theorems? And, yeah, I don't know who wants to go first. But, yeah, what's your favorite theorem?

KK: Yeah, let’s hear it.

PH: I’ll do it. I’ll go first. I always like hearing Aris talk. So I'm just like “Aris, go,” right? But no, I’m going to take the lead today. Alright, so I wanted to tell you about this theorem called Zeckendorf’s theorem. I don't know if you know about it.

KK: I do not.

PH: And it goes like this. So start with the Fibonacci numbers without the repeated 1. So 1, 2, and then start adding the previous two, so 3-5-8, and so on. Alright. So if you start with that sequence, his theorem says the following, if you give me any positive integer N, I can write it uniquely as a sum of non-consecutive Fibonacci numbers.

AW: Oh, wow!

EL: Uniquely?

PH: Yes. And this is why you need to get rid of the 1, 1. Because otherwise you have a choice. But yeah. So it's hard to do off the top of my head, because I'm not someone who, like, holds numbers. But say, for example, we wanted to do 20. Maybe we wanted to write the number 20 as a sum of Fibonacci numbers that are not consecutive. So what would you do? You would find the largest Fibonacci number that fits inside of 20. So in this case, it would be 13.

AW: Yeah.

PH: 13 fits in there. Okay, so we subtract 13. We're left with 7. Repeat the pattern.

KK: Ah, five and two.

KK: Five and two! They're non consecutive.

KK: Okay.

PH: Yeah.

AW: Wow!

PH: Three is in between them, and eight is in between the others. And so you can do this uniquely. And so this is using what's known as the greedy algorithm because you just do that process that I just said, and it terminates because you started with a finite number.

KK: Sure.

PH: And so the the proof, of course, there's the, you know, “Can you do it?” but then “Can you do it uniquely?” So the thing that you would do there is assume that you have two different ways of writing it, each of which uses non-consecutive, and then you would argue that they end up being exactly the same thing. So that, in fact, they use the same number of Fibonacci numbers and that those numbers are actually the exact same.

KK: Sure, okay.

EL: Yeah. Like I'm trying to figure out — and I don't, I also am not super great at working with numbers in my head just on the fly. But yeah, I'm trying to figure out, like, what would have gone wrong if I had picked eight instead of 13 to start with, or something? And I feel like that will help me understand, but I probably need to go sit quietly by myself and think about it. Because there’s a little pressure.

PH: Yeah, it's a little subtle. And it might be that you don't get big enough, you end up having to repeat something.

EL: Yeah, I feel like there's not enough left below eight to get me there without being consecutive.

PH: Yeah. Right.

AW: Right. Because you’ve got to get 12. Yeah, yes. Yeah.

KK: Yeah, it makes sense, right? Like, I guess, you know, if you pick the largest one less than your number, then it's more than halfway there. That's sort of the point, right? So that's how you prove it terminates, but also the the non-uniqueness, the non-uniqueness seems like the hard part to me somehow, but also the non-consecutive. Wait a minute, I don't know, which is.

AW: Well it sounds hard, period.

KK: Yeah. I like this theorem. This is good. What attracts you about this theorem? What gets you there?

PH: So I, in part of my dissertation, I found a new place where the Fibonacci numbers showed up. And so once you find Fibonacci numbers somewhere new, I was like, what else is known about these beautiful numbers? And so this was one of those results that I found, you know, just kind of looking at the literature. And then I later on started doing some research generalizing this theorem. So meaning, in what other ways could you create a sequence of numbers that allows you to uniquely write any positive integer in this kind of flavor, right, that you don't use things consecutively, and consecutively, really, in quotes, because you can define that differently. And so it led me to new avenues of research that then I got to do. It was the first few research projects with some of my undergraduate students at the Military Academy. And then I learned through them — they looked him up — that he actually came up with this theorem while he was a prisoner of war.

AW: Oh, wow!

PH: This is when Zeckendorf worked on this theorem. And to me, this was really surprising that, you know, my students found this out. And then I was like, “See, mathematics, you can just take it anywhere.” lLke this poor man was a prisoner of war, and he's proving a theorem in his cell.

KK: Well Jean Leray figured out spectral sequences in a German POW camp.

PH: I did not know that!

AW: Anything to pass the time.

KK: What else are you going to do?

EL: I mean, Messiaen composed the Quartet for the End of Time — I was about to say string quartet, but it's a quartet for a slightly different instrumentation, in a concentration camp, or a work camp. I'm not sure. But yeah, I'm always amazed that people who can do that kind of creative work in those environments, because I feel like, you know, I've been stuck in my house because of a pandemic, and I'm, like, falling apart. And my house is very comfortable. I have a comfortable life. I am not as resilient as people who are doing this. But yeah, that is such a cool theorem. I'm so glad that you said that. And I'm trying to think, like, Lucas numbers are another number sequence that are kind of built this way. And so is there anything that you can tell us about the the sequences that you were looking at, like, I don't know, does this work for Lucas numbers? I don't know if you've looked at that specifically, or did you look at ways to build sequences that would do this?

PH: Yeah. So we started from the construction point of view. So rather than give me a sequence, and then tell me how you can uniquely decompose a number into a sum of elements in that sequence, we worked backwards. So one of the research projects that we started with is what we called — there's a few of them — but one, it was a “Generacci” sequence. And so what we would do is, instead of thinking of the numbers themselves, imagine that you have buckets, an infinite number of buckets, you know, starting out the first bucket all the way to infinity, and you get to put numbers into the sequence in the following way. So you input the number 1 to begin with, because you need a number to start the sequence. And since you want to write all positive integers, well, you’ve got to start with 1 somewhere. So you stick the number 1 in the first bucket. And then you set up some system of rules for which buckets you can use to pull numbers from that then you add together to create new numbers. Well, you only have one bucket, and you only put the number 1 in it. So then you move to the next bucket. Well, okay, you want to build the number two, and you only have the number one, and as soon as you pull it from the bucket, you don't have any other numbers to use. So let's stick the number two in the second bucket. Oh, well, now I could maybe in my rule, grab a number from two buckets, and add them together to get the next number. Oh, that starts looking familiar. The third bucket will have not the number three, because you were able to build it. So what next number could you grab? Well, maybe you can stick in the 4 in there. And so by thinking of buckets, the numbers that you can fill the buckets with, that you couldn't create from grabbing numbers out of previous buckets under certain rules, you now start constructing a sequence. And provided that you very meticulously set up the rules under which you can grab numbers out of the buckets to add together to build new numbers, then you do not need to add that number into the buckets, because you've already built it.

EL: So what rules you have about the buckets will determine what goes in the buckets.

PH: Exactly, exactly. So you might say okay, maybe our buckets can contain three numbers. And you're not allowed to take numbers out of consecutive buckets, or neighboring buckets, or you must give five buckets in between. So what must go into the buckets to guarantee that you can create every single number and you can do so only uniquely? And so these are these bin decompositions of numbers. But you are working backwards. You start with all the numbers, and then you decide how you can place them in the buckets and how you can pull them from the buckets to add together. So I'm being vague on purpose, because it depends on the rules. And actually it's quite an open area of research, how do you build these sequences? You set up some some capacity to your bucket, some rules from where you can pull to add together. And the nice thing is that it's very accessible, and then it leads to really beautiful generalizations of these kinds of results like that of Zeckendorf.

EL: This is very cool.

AW: Fantastic.

EL: All right. So Aris, I feel like the gauntlet has been thrown.

KK: Yeah.

AW: Yeah, well mine is simple. This is not a competition. Yeah, no. I guess mine is influenced — I’ve been thinking about a bunch of different things, but I keep coming back to the same one, which I think is influenced by my identity as a teacher first and foremost when I think about the fundamental theorem of calculus. I just keep coming back to that one. And I don't know how many people have used this one with you on this podcast before, but for me, it hits so many of the check marks of my identity in terms of thinking about myself as a mathematician and a teacher, in the sense that for a lot of students who get to calculus, it's one of the first major, major theorems that will show up in their faces that we actually call out and say this is a theorem. And we call it fundamental, right? We don't often bring up the fundamental theorem of algebra in college algebra, right? Or in other places, or the fundamental theorem of arithmetic, right? But so it's one of these first fundamental theorems. And so it also helps to tell the story of a course, right? And so that really hits the teacher part of me where too often people in the calculus sequence spend all this time talking about derivatives, and all of a sudden, we just switch the anti-derivatives. And we don't really say why. You'll figure out in the next couple of sections, and then we start adding rectangles, and we don't say why. And so it really is, at least the way that the order of calculus has gone and in terms of how to teach it, in my experience, it really is this combination, like, oh, this is why we've been doing this. And this is the genius of relating two things. Sometimes I've gone in, and I've talked about, like I put up a sine curve, and a cosine curve, and we talk about how one of them measures the area under the curve. And then I pretend to bump my head and get amnesia. And then I'll come and say, “Oh, look, looks like we've been talking about derivatives. Right?” And they’re like, “Wait, what do you mean we're talking about derivatives?” “This is the derivative of this one.” And they’ll go, “What? We were measuring the area under the curve.” “Well, we’re also measuring the derivative, right?” This is the derivative, but this is measuring the area. And it's like, Oh, right, and so it's just one of these “aha” moments, where if people have been paying attention, it's like, oh, that's actually pretty cool, right? And then also in terms of the subject itself in relationship to high school, just really thinking about — because I get a lot of students who know all the rules, right? And they look at the anti-derivative with the integral sign and say “That's the integral.” Well, that's an anti-derivative, right?

EL: We’re not there yet.

AW: Yeah, that the anti-derivative and the integral are actually different. And so just having that conversation. And it also is a place to talk about the history of the subject and stuff like this.

EL: Yeah, I love it. And, at least for me, I feel like it's a slow burn kind of theorem. The first time you see it, you're like, “Okay, it's called the fundamental theorem of calculus. I guess some people think it's really important.” So that might be your Calculus I class. And then you see it again, maybe in an introductory real analysis class. And you're like, “Oh, there's more here.” And then you teach calculus, and you’re like, “Ohhhh!”

AW: Oh right, yes!

EL: Your brain explodes. You're like, “This is so cool!” And then your students are where you were several steps ago. And they’re like, “Okay, I guess it’s all right.”

AW: If I get the success rate of like, I've had three or four people go, “Whoa!” And it's like, okay, yeah, you're with me. And so this is out of hundreds of people.

EL: If you can get a few people that do that the first time they see it, that’s awesome.

AW: Yes. Yeah. No, it's been fulfilling for sure. And so then the proof itself, you know, it's also great, because then it culminates all of the theorems that you've been talking about beforehand. Depending on the proof, of course, but like, there's the intermediate value thereorem. There's the mean value theorem for integrals. There's unique continuity, at least in this version of it, in order for it to work. So yeah, it's great.

KK: So when you teach calculus, there's always two parts to the fundamental theorem. And so I like the one where the derivative of the integral is the function back, right? That's the fun, like for the mathematician in me, this is the fun part. Your students never remember that. Right? They always remember the other one, where we evaluate definite integrals by finding it the anti-derivative. So I was going to ask, if you had to pick one of the two, which one is your favorite?

AW: I mean, part of it is because at least the way that I've taught it, we're coming out of the mire of Riemann sums.

KK: Right.

AW: And so people have suffered through doing rectangles so much. And then I just get to say, “Oh, you don't have to do this anymore.” I mean, I've had a few students go, you know, now that we do — I always use the antiderivative of x squared on zero to 10, or the area under the curve of x squared from zero to 10. And like, sometimes I'll say, “Oh, that's 1000 over 3, right?” And then it was like, “Well, how did you do that so quickly?” We'll see. Right? But then, at the end, when I'll say, okay, and then we do another one again. And then I show how to apply the theorem, and people say, “Well, why didn't you just say that?” And then we have a great conversation there about how this isn't about the answer, that this is about a process and understanding the impact of mathematical ideas, that the theorem, as with all theorems, but this one is my favorite, is an expression of deep human intellect. And that if we reframe what theorems are, we get a chance to rehumanize mathematics. And so I think that too often in our math classes, and our math discourse, we remove the theorems from the humanity of the people who created them. And so people get deified, like Newton and Leibniz, but you know, these same people had to sit down and work hard at it and figure it out.

KK: No, it’s certainly a classic, but it is surprising how little it has come up on our podcast. It was the very first episode.

AW: Oh, okay.

KK: Yeah. Amie Wilkinson chose it. And then this will be episode 60-something.

AW: Okay. Yeah.

PH: Wow.

EL: We've talked, we've mentioned it in some other episodes. But it isn’t — I mean, there are just — I love this podcast, obviously, I keep doing it. And there are just so many types of theorems. And I love that you two picked different types. Yours, Aris, is one of these classics. Everyone who gets to a certain point in math has seen it, hopefully has appreciated it also. And Pamela, you picked one that none of us had ever heard of and made us say, “Whoa, that's so cool!” And people just have so many relationships with yours. And that's what this podcast is really about. Actually it's not about theorems. It's about human relationships with theorems and what makes humans enjoy these theorems. And so you picked two different ways that we enjoy theorems. And I just love that. So yeah.

KK: Yeah, that is what we're about here, actually. I mean, I mean, yeah, the theorem. But actually what I like most about our podcast, so let's toot our own horn here. We’re trying to humanize mathematics. I think everybody has this idea that mathematicians are a very monolithic bunch of weird people who just — well, in movies we’re always portrayed as either being insane, or just completely antisocial. And I mean, there’s some truth and every stereotype, I suppose, but we are people, and we love this thing. We think it's so cool. And sharing that with everyone is really what's so much fun.

AW: Yeah. And I think also that, for me, the theorem itself, and what it reveals, touches something that’s inside of us. There’s something about it, right? There’s the “Whoa” part that is that is indescribable and that I think really touches to our humanity. There is a eureka moment where you're just like, “Oh, I understand this now.” Or this connection is amazing, right? Yeah, it's indescribable.

KK: So we all agree these things are beautiful. So here's a question. Where do people lose this? I mean, I have a theory, but — because we've all had this experience, right? You're at a cocktail party and someone says, they find out you're a mathematician and like, oh, record scratch. I hate math. Okay.

AW: Yes, yes, yes.

PH: But I don’t think they hate math, though, Kevin.

KK: No, they don’t. Nobody hates math. Nobody hates math when they're a kid. That's exactly right. So I think when they say that they mean that the algebra caused them trouble. When x’s started showing up.

PH: I don't even think that's it.

KK: Okay. Good. Enlighten me because I want an answer to this that I can’t find.

PH: I don't think it's that people hate math or that they hate that the alphabet showed up all of a sudden in math that they hate how people have made them feel when they struggle with math. Math is an inanimate object. Math is not going out there and, like, punching people in the face. It's the way that people react to other people's math. Right? The second that you don't use the language in the way that somebody expects you to use it and you're trying to communicate properly and somebody says, “That’s not how you say it. It's not FOILing. It's called distributing!” Right? But you knew what I meant when I said FOIL the binomial!

KK: Of course I did.

PH: FOILing this gives you the middle term, blah, blah, right? So it’s again about human interactions. And if you make someone feel dumb, they'll never like what it is that they're trying to learn

AW: Amen to that. And they will conflate the two, which is what always happens.

PH: That’s exactly it!

AW: They will replace the experience with the subject itself, when in fact, they're talking about the experience. Yeah. So yeah, we've been working a lot about this in the last few years, Pamela and I and Dr. Michael Young, about when people say they hate mathematics, they’re really talking about their mathematical experience. So my immediate response to your question is just bad teaching. Let's just call it what it is.

PH: Right.

AW: I don't want to get on my podcast too early. We're recording later.

PH: We’re recording in a bit, yeah.

AW: But yeah, we're talking about people. And I say this as a loving critique of the greatest discipline in the history of people. I truly believe that, but I believe that the way we teach it, and the cultural norms we take with it, devalues people, and so I want every person who's listening to this now to then the next time they hear somebody say they hate it, look at them as an innocent person who had a bad mathematical experience. And then, because I see too often amongst my people in the community who say they hate having these conversations with people who say they hate it. And I think we need to return innocence back to that person. And say that this is not a person who hates you or even hates the subject. This is a hurt person. Yes, this is a person who has been damaged in our subject. And by the way, I go much farther than that. It's our responsibility to try and help repair that because this person is going to impact their cousin, their child, their relative, by bringing this hate of the subject, when in fact, it doesn't have anything to do with the subject.

EL: Yeah. It’s about the traumatic experiences. And actually, I think mathematicians often have a bit of a persecution complex and think this is the only place where people have this reaction. But one of my hobbies is singing, and in particular, singing with large groups of untrained people who are just singing because we love singing. And the baggage that people bring to singing is similar. I’m not saying it's entirely the same, but people have been made to feel like their voice isn't good enough.

AW: Yes.

EL: They have this trauma associated with trying to go out and do this sometimes. Obviously a lot of people love to sing and will do it in public. A lot of people love to sing at home and are scared of doing it in public because they're worried about, you know, their fourth grade music teacher, who told them to sing quieter, or whatever happened.

PH: Yes,

AW: That’s right. That's right. And the connection is similar, because what are we saying? We're saying that if you don't hit this right note, then it doesn't count. As opposed to if you don't get the answer seven, then we're not going to value you because the answer is seven, right? Because we have this obsession with the correct answer in mathematics. Right.

PH: And not only that, but also doing it fast.

AW: Yes.

PH: You and I have talked about this before, that — maybe in singing, this is different. I'm not sure. I definitely can relate to the trauma of never singing out loud in public. But is there this same sentiment that you must get it perfect the first time and pretend that it doesn't actually take you hours of training?

EL: I mean, it comes up. There’s definitely, people can feel more valued if they're quicker at picking things up than others, although, you know, it's not the same. There's no isomorphism between these two, I don't know, to bring a little silly math lingo in. But there definitely, there are a lot of similarities, and I think about this a lot, because two things I love in my life are math and singing with my friends. And, you know, I just see these relationships. But yeah, I could go on a whole rant, and I want to not do that.

AW: No, no, no, I appreciate you bringing it up.

EL: But I think it's a really interesting correspondence.

AW: And then the final one is that, you know, in the music space, what is it that we really should be trying to do, value everybody's voice? And in mathematics, we should be valuing everybody's contribution. Right? This is all we're saying. And what does each discipline look like when we value people's voices, no matter where they are on the keys? And we value everyone's contribution to trying to solve a problem.

EL: Yeah, yeah. And how can we help people, you know, grow in the way they want to? You can say, like, “Oh, I like I am not as good a sight reader as I want to be. How can I get better?” How can we help people grow in that way without feeling cut down?

AW: Yeah.

EL: Yeah, it is true for math, too. Yeah. It's just, everything is connected. Woo.

AW: Yes. But you know, we've been talking about, you know, these human relationships we all have with math. And so another part of our podcast that we love is forcing you to do make one more human connection between math and something else with the pairing. So what goes well, Pamela, with this theorem about uniquely writing the numbers in terms of the Fibonacci sequence?

PH: So I was trying to think about my favorite food, and when it was the epitome of perfection, and I came up with, okay, so if we're going to pair it with something to drink, I was like, I want to think about happy moments. Because this feels like a happy theorem. And so I want to go with some champagne.

KK: Okay.

PH: Okay, I was like, “We're gonna go fancy with it!” But then for food, I'm thinking about, oh, this is hilarious. So I went to a conference in Colombia, we visited Tayrona which is a beach in Colombia. And on the side of the beach, I paid to have ceviche, fresh ceviche. And I've never been happier eating anything in my life. And so I imagine myself learning Zeckendorf’s theorem at the beach in Tayrona in Colombia, with some champagne and the ceviche.

EL: Oh man.

AW: Wow.

PH: Beat that, Aris! Beat. That.

AW: There’s no way. So wait, so I want to make sure I understand. So is this while you're reading the proof? Or is this while you’re—

PH: This is like the gold standard. If I were to put all the, like, uniqueness of my favorite food, my favorite drink and my favorite theorem, I would put them in a location which is Tayrona in Colombia, at the beach, eating ceviche sipping on some champagne, learning Zeckendorf’s theorem.

AW: Okay.

KK: Is this the Pacific coast or the Caribbean?

PH: You’re asking questions I should know the answer to, and I believe it’s the Caribbean.

KK: Okay.

PH: Nobody Google that. [Editor’s note: I Googled that. It is the Caribbean.] I have no idea where they took me in Colombia. I just went.

KK: Sure.

EL: Yeah, that sounds so lovely as I look out of my window where there's snow and mud from some melted snow.

PH: Ditto.

AW: So I yeah, I think for the fundamental theorem of calculus, I think this is something that's just classic. Like you're just having a nice pizza and some ginger ale. You're just sitting down and you're enjoying something hopefully that everybody likes and that connects with everybody, that everybody hopefully sees that they get to get that far. So yeah, I mean, my daughter recently — I didn't realize this. She's 9. And we were talking. We visited my aunt in DC. My aunt raised me. And my daughter was much younger at that time, but then every time she thinks about going to visit, she thinks about the ginger ale that my aunt got her because that was the only time she ever got ginger ale. So she’s like, “Oh, I like your aunt, Daddy, because you know, I had ginger ale there.” And I was like, Oh, I should have ginger ale more often. So that made me think of that.

PH: That’s adorable.

EL: I can really relate to that feeling of, like, when you're a kid, something that is totally normal for someone else isn't what's normal for your family. So you think it's a super special thing.

AW: It’s amazing.

EL: I think I had this with, like, Rice-a-Roni or something at my aunt's house, and my mom didn't use Rice-a-Roni, and I was like, “Whoa, Mom, you should see if you can find Rice-a-Roni.”

PH: Amazing.

EL: She was like, “Yeah, they have Rice-a-Roni here.”

AW: Rice-a-Roni’s the best.

KK: I haven't had that in years. I should go get some.

AW: Me either. All right.

PH: That’s how you know you made it.

KK: You know what? You know, single mom and all that, and I lived on Kraft macaroni and cheese when I was a kid. And yeah, you would think I don't like it any more. But, aw man.

PH: Listen, that thing is delicious. So good.

AW: I was about to say.

EL: They know what they’re doing. Yeah. Well, that's great. And I mean, pizza is my favorite food. As great as ceviche on the beach sounds, pizza, just, when you come down to it, it's my favorite food. And so I love that you paired the fundamental theorem of calculus with my favorite food.

KK: So I'm curious, there must be a human who doesn't like pizza, but have you ever met one? I've never met one.

PH: No.

EL: I know people who don't like cheese. And cheese is not — I mean, to me cheese is essential to the pizza experience, but you can definitely do a pizza without cheese.

AW: Yeah. No, my wife also always says that for her it's about the sauce. So I think she might be a person who can get rid of the cheese if the sauce is right. Yeah.

KK: But the crust better be good too.

AW: Of course, of course. It's a full package here.

EL: But okay, so you say that, but on the other hand, I would say that bad pizza is still really good.

KK: Sure.

EL: I mean, you can have pizza that you're like, “I wish I didn't eat that.” But I have very rarely in my life encountered a slice of pizza that was like, “Oh, I wish I wish I had done something else other than eat that pizza.”

AW: It’s actually a pretty unbeatable combination, right? Tomato sauce, cheese and bread.

PH: Yeah. It kind of can't go wrong. Yeah.

KK: When I was when I was in college, there was a place in town. It was called Crusty’s Pizza, and I don't think it exists anymore. And it was decidedly awful. But we still got it because it was cheap. So we would occasionally splurge on the good pizza. But you could get a Crusty’s pie for like five bucks.

AW: Absolutely.

KK: This is dating myself. But yeah, absolutely. Always. All right, so we've got we've got theorems, we’ve got pairings. You've plugged your podcast pretty well, although you can talk about it more if you'd like. Anything else that either of you want to plug, websites, the Twitter?

EL: Yeah, but can you say a little more about the book that you mentioned?

AW: Yeah, the book is a series of dialogues that was an extension of an AMS webinar series that we gave about advocating for students of color mathematics. And so we had just decided, you know, there was so much momentum, we had hundreds of people coming every time to the four-part series. And so we were like, you know, we've gotten to a place where we've given all these talks, and then you give talks, create momentum, and then it just ends. And we're just like, you know, what, not this time. Let's create a product out of this. And so, we decided quickly to get the book together, just answering some of the unanswered questions from the webinar series. So we had the motivation, in terms of answering their questions. And yeah, we got it together. And it was an honor. So it really is just a list of our dialogues, a transcription of our dialogues, answering some of the unanswered questions from that webinar series. And so it's gotten some really good reviews, and people are using it in their departments. And so it's been fantastic so far.

PH: Yeah, I think that's that's the part that I'm really enjoying, getting the emails from people who have purchased the book. And so maybe I should say the full title, so it is Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics. And the things that I hear from folks who have purchased the book — so thank you all so much for the support — is that they didn't expect that there is part of a workbook involved in the book. So it isn't just Aris and I going back and forth at telling you things. I mean, a lot of that there is, that is part of the content. But there's also a piece about doing some pre-reflection before we start hearing some of the dialogue that we have, and then also the post part of it. So how are you going to change? And how are you going to be a better advocate for students of color in mathematics? And so it leaves the reader with really a set of tools to come back to time and time again. That's really what I see as a benefit of the book. And people are purchasing it as a department to actually hold some kind of book club and really think about what of the things that we suggest that professors implement in their department, in their classrooms, in their institutions, what they can actually do. And so the reception has been really wonderful. And I'm just super thankful that people purchase the book, and we're supporting our future work.

EL: Yeah. And can you also mention, is it minoritymath.org, the website that hosts Mathematically Uncensored?

AW: That’s correct. That's right. So yeah, that's the home of the podcast. And that's a place where we're trying to create voices for underrepresented minorities in the mathematical sciences. And so you can go there not just for the podcast, but for other content as well that centers around that experience.

KK: Okay.

EL: Fantastic. Thank you so much for joining us.

KK: Yeah.

EL: I had a blast.

PH: Thank you.

KK: This was a really good time.

EL: Yeah. Over lunch today, I'm going to be writing down numbers and writing them in terms of Fibonacci numbers. It’s great.

AW: It will be fantastic.

PH: Awesome.

AW: Thanks.

PH: Bye, everyone.

KK: Thanks, guys.

On this very special episode, we had not one but two guests, Pamela Harris from Williams College and Aris Winger from Georgia Gwinnett College, to talk about their podcast, Mathematically Uncensored, and of course their favorite theorems. Here are some links you might be interested in as you listen to the episode.

Winger's profile on Mathematically Gifted and Black

Mathematically Uncensored, the podcast they cohost

Minoritymath.org, the Center for Minorities in the Mathematical Science, a website with information and resources for people of color in mathematics

Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics, their book

Zeckendorf's theorem and a biography of Edouard Zeckendorf

Jean Leray, a French mathematician who worked on spectral sequences as a prisoner of war

Olivier Messiaen's Quartet for the End of Time, composed when he was a prisoner of war

A paper generalizing the Zeckendorf theorem by Harris and coauthors

Our episode with Amie Wilkinson, who also chose the Fundamental Theorem of Calculus, making it 2 for 2 among mathematicians with the initials AW.

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